Finite Group Databasebeta

← C5×#85486 #14511224

Subgroups — C5×#85486 #14511224

Order 1920.

994 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C5×A4C211yesmaximal, characteristic, complemented
2C2×C5×GL(2,ℤ/4)C212yesmaximal, complemented
3C2×C5×C4⋊S4C211yesmaximal, characteristic, complemented
4C5×(C2×S4)⋊C4C212yesmaximal, complemented
5C5×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6C5×#2847—31nomaximal, Hall
7C23×C5×A4C2211yescharacteristic
8C22×C5×S4C2212yes—
9C2×C4×C5×A4—22no—
10C2×C4×C5×A4C2211yescharacteristic
11C2×C4×C5×A4C2211yescharacteristic
12C5×GL(2,ℤ/4)—24no—
13C2×C5×SL(2,ℤ/4)C2212yes—
14C5×C4⋊S4—22no—
15C5×#930—41nomaximal
16#85486C511yesmaximal, characteristic, direct product factor, Hall
17C24×C4×C5D311yescharacteristic, complemented
18C2×C5×#121—31no—
19C2×C5×#121—31no—
20C2×C5×#121—31no—
21C2×C5×#120—32no—
22C5×#386—32no—
23C5×#386—32no—
24C5×#386—32no—
25C5×#380—31no—
26C5×#380—31no—
27C5×#380—31no—
28C22×C5×A4—21no—
29C22×C5×A4—21no—
30C22×C5×A4D412yescomplemented
31C22×C5×A4C2311yescharacteristic
32C2×C5×S4—22no—
33C2×C5×S4—22no—
34C2×C5×S4—22no—
35C22×C3×C4×C5—41no—
36C2×C5×#67—42no—
37C2×C5×D12—41no—
38C4×C5×A4—21no—
39C4×C5×A4D412yescomplemented
40C5×SL(2,ℤ/4)—22no—
41C5×(C2×D3)⋊C4—42no—
42C5×C4⋊Dic3—41no—
43C22×C4×A4C2×C511yescharacteristic, complemented
44C2×GL(2,ℤ/4)C2×C512yescomplemented
45C2×C4⋊S4C2×C511yescharacteristic, complemented
46(C2×S4)⋊C4C2×C512yescomplemented
47C4⋊SL(2,ℤ/4)C2×C511yescharacteristic, complemented
48C25×C5C2×D311yescharacteristic
49C22×C5×D4—32no—
50C22×C5×D4—32no—
51C22×C5×D4—31no—
52C22×C5×D4—32no—
53C23×C4×C5—22no—
54C23×C4×C5—31no—
55C23×C4×C5—31no—
56C23×C4×C5—31no—
57C23×C4×C5—31no—
58C23×C4×C5—62no—
59C23×C4×C5C2×D311yescharacteristic
60C23×C4×C5C2×D311yescharacteristic
61C5×#121—62no—
62C5×#121—64no—
63C5×#121—61no—
64C5×#121—61no—
65C5×#121—61no—
66C5×#121—62no—
67C5×#121—61no—
68C5×#120—64no—
69C5×#120—64no—
70C2×C5×C4⋊C4—31no—
71C2×C5×C4⋊C4—31no—
72C2×C5×C4⋊C4—31no—
73C2×C5×C22⋊C4—32no—
74C2×C5×C22⋊C4—32no—
75C2×C5×C22⋊C4—32no—
76C2×C5×C22⋊C4—32no—
77C2×C5×C22⋊C4—32no—
78C2×C5×C22⋊C4—32no—
79C5×#95—32no—
80C5×#95—32no—
81C5×#95—32no—
82#2847—31noSylow
83C23×C3×C5—41no—
84C22×C5×D3—42no—
85C2×C5×A4—22no—
86C2×C5×A4C2×D411yescharacteristic
87C2×C5×A4C2×D411yescharacteristic
88C2×C5×A4#4111yescharacteristic
89C5×S4—42no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5—41no—
92C2×C3×C4×C5—82no—
93C5×#67—84no—
94C2×C5×Dic3—42no—
95C5×D12—82no—
96C23×A4C22×C511yescharacteristic
97C22×S4C22×C512yes—
98C2×C4×A4—22no—
99C2×C4×A4C22×C511yescharacteristic
100C2×C4×A4C22×C511yescharacteristic
101GL(2,ℤ/4)—24no—
102C2×SL(2,ℤ/4)C22×C512yes—
103C4⋊S4—22no—
104#930—41no—
105C24×C5—21no—
106C24×C5—21no—
107C24×C5—31no—
108C24×C5—31no—
109C24×C5—32no—
110C24×C5—32no—
111C24×C5—61no—
112C24×C5—61no—
113C24×C5D1212yescomplemented
114C24×C5C22×D311yescharacteristic
115C2×C5×D4—61no—
116C2×C5×D4—62no—
117C2×C5×D4—62no—
118C2×C5×D4—64no—
119C2×C5×D4—62no—
120C2×C5×D4—64no—
121C2×C5×D4—62no—
122C2×C5×D4—64no—
123C2×C5×D4—62no—
124C2×C5×D4—62no—
125C2×C5×D4—64no—
126C2×C5×D4—64no—
127C2×C5×D4—62no—
128C2×C5×D4—62no—
129C2×C5×D4—62no—
130C2×C5×D4—62no—
131C2×C5×D4—61no—
132C22×C4×C5—21no—
133C22×C4×C5—31no—
134C22×C4×C5—31no—
135C22×C4×C5—31no—
136C22×C4×C5—32no—
137C22×C4×C5—34no—
138C22×C4×C5—31no—
139C22×C4×C5—31no—
140C22×C4×C5—31no—
141C22×C4×C5—31no—
142C22×C4×C5—32no—
143C22×C4×C5—32no—
144C22×C4×C5—32no—
145C22×C4×C5—62no—
146C22×C4×C5—62no—
147C22×C4×C5—61no—
148C22×C4×C5—61no—
149C22×C4×C5—62no—
150C22×C4×C5—62no—
151C22×C4×C5—62no—
152C22×C4×C5—62no—
153C22×C4×C5—62no—
154C22×C4×C5#6712yescomplemented
155C22×C4×C5S411yescharacteristic, complemented
156C5×C4⋊C4—61no—
157C5×C4⋊C4—62no—
158C5×C4⋊C4—62no—
159C5×C4⋊C4—61no—
160C5×C22⋊C4—64no—
161C5×C22⋊C4—64no—
162C5×C22⋊C4—62no—
163C5×C22⋊C4—64no—
164C5×C22⋊C4—62no—
165C5×C22⋊C4—64no—
166C5×C22⋊C4—62no—
167C5×C22⋊C4—62no—
168C24×C4C5×D311yescharacteristic, complemented
169C2×#121—31no—
170C2×#121—31no—
171C2×#121—31no—
172C2×#120—32no—
173#386—32no—
174#386—32no—
175#386—32no—
176#380—31no—
177#380—31no—
178#380—31no—
179C22×C3×C5—42no—
180C22×C3×C5—41no—
181C22×C3×C5—81no—
182C22×C3×C5—81no—
183C2×C5×D3—82no—
184C2×C5×D3—82no—
185C2×C5×D3—82no—
186C5×A4#12111yescharacteristic, complemented
187C3×C4×C5—42no—
188C3×C4×C5—81no—
189C5×Dic3—82no—
190C22×A4—21no—
191C22×A4—21no—
192C22×A4C5×D412yescomplemented
193C22×A4C23×C511yescharacteristic
194C2×S4—22no—
195C2×S4—22no—
196C2×S4—22no—
197C22×C3×C4—41no—
198C2×#67—42no—
199C2×D12—41no—
200C4×A4—21no—
201C4×A4C5×D412yescomplemented
202SL(2,ℤ/4)—22no—
203(C2×D3)⋊C4—42no—
204C4⋊Dic3—41no—
205C23×C5—22no—
206C23×C5—31no—
207C23×C5—32no—
208C23×C5—31no—
209C23×C5—31no—
210C23×C5—31no—
211C23×C5—32no—
212C23×C5—32no—
213C23×C5—32no—
214C23×C5—32no—
215C23×C5—31no—
216C23×C5—61no—
217C23×C5—61no—
218C23×C5—61no—
219C23×C5—61no—
220C23×C5—62no—
221C23×C5—61no—
222C23×C5—61no—
223C23×C5—62no—
224C23×C5—62no—
225C23×C5—61no—
226C23×C5—64no—
227C23×C5—62no—
228C23×C5—62no—
229C23×C5—62no—
230C23×C5—62no—
231C23×C5—62no—
232C23×C5—62no—
233C23×C5—62no—
234C23×C5C2×D1211yescharacteristic
235C23×C5#23511yescharacteristic
236C23×C5C2×#6711yescharacteristic
237C23×C5C2×S411yescharacteristic
238C5×D4—122no—
239C5×D4—122no—
240C5×D4—122no—
241C5×D4—122no—
242C5×D4—128no—
243C5×D4—124no—
244C5×D4—122no—
245C5×D4—124no—
246C5×D4—122no—
247C2×C4×C5—22no—
248C2×C4×C5—31no—
249C2×C4×C5—32no—
250C2×C4×C5—32no—
251C2×C4×C5—32no—
252C2×C4×C5—32no—
253C2×C4×C5—31no—
254C2×C4×C5—62no—
255C2×C4×C5—61no—
256C2×C4×C5—61no—
257C2×C4×C5—61no—
258C2×C4×C5—62no—
259C2×C4×C5—64no—
260C2×C4×C5—62no—
261C2×C4×C5—61no—
262C2×C4×C5—62no—
263C2×C4×C5—62no—
264C2×C4×C5—62no—
265C2×C4×C5—62no—
266C2×C4×C5—62no—
267C2×C4×C5—62no—
268C2×C4×C5—62no—
269C2×C4×C5—62no—
270C2×C4×C5—62no—
271C2×C4×C5C2×S411yescharacteristic
272C2×C4×C5C2×S411yescharacteristic
273C25C2×C5×D311yescharacteristic
274C22×D4—32no—
275C22×D4—32no—
276C22×D4—31no—
277C22×D4—32no—
278C23×C4—22no—
279C23×C4—31no—
280C23×C4—31no—
281C23×C4—31no—
282C23×C4—31no—
283C23×C4—62no—
284C23×C4C2×C5×D311yescharacteristic
285C23×C4C2×C5×D311yescharacteristic
286#121—62no—
287#121—64no—
288#121—61no—
289#121—61no—
290#121—61no—
291#121—62no—
292#121—61no—
293#120—64no—
294#120—64no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C22⋊C4—32no—
299C2×C22⋊C4—32no—
300C2×C22⋊C4—32no—
301C2×C22⋊C4—32no—
302C2×C22⋊C4—32no—
303C2×C22⋊C4—32no—
304#95—32no—
305#95—32no—
306#95—32no—
307C2×C3×C5—41no—
308C2×C3×C5—41no—
309C2×C3×C5—41no—
310C2×C3×C5—82no—
311C5×D3—162no—
312C23×C3—41no—
313C22×D3—42no—
314C2×A4—22no—
315C2×A4C2×C5×D411yescharacteristic
316C2×A4C2×C5×D411yescharacteristic
317C2×A4C5×#4111yescharacteristic
318S4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—82no—
322#67—84no—
323C2×Dic3—42no—
324D12—82no—
325C22×C5—21no—
326C22×C5—21no—
327C22×C5—31no—
328C22×C5—31no—
329C22×C5—31no—
330C22×C5—31no—
331C22×C5—31no—
332C22×C5—31no—
333C22×C5—31no—
334C22×C5—31no—
335C22×C5—31no—
336C22×C5—34no—
337C22×C5—32no—
338C22×C5—62no—
339C22×C5—62no—
340C22×C5—62no—
341C22×C5—62no—
342C22×C5—61no—
343C22×C5—62no—
344C22×C5—62no—
345C22×C5—62no—
346C22×C5—61no—
347C22×C5—61no—
348C22×C5—122no—
349C22×C5—122no—
350C22×C5—122no—
351C22×C5—122no—
352C22×C5—122no—
353C22×C5—122no—
354C22×C5—122no—
355C22×C5#93011yescharacteristic, complemented
356C22×C5C4⋊S412yescomplemented
357C22×C5C22×S411yescharacteristic
358C4×C5—21no—
359C4×C5—32no—
360C4×C5—61no—
361C4×C5—122no—
362C4×C5—122no—
363C4×C5—122no—
364C4×C5GL(2,ℤ/4)12yescomplemented
365C24—21no—
366C24—21no—
367C24—31no—
368C24—31no—
369C24—32no—
370C24—32no—
371C24—61no—
372C24—61no—
373C24C5×D1212yescomplemented
374C24C22×C5×D311yescharacteristic
375C2×D4—61no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—64no—
379C2×D4—62no—
380C2×D4—64no—
381C2×D4—62no—
382C2×D4—64no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—64no—
386C2×D4—64no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—61no—
392C22×C4—21no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—32no—
397C22×C4—34no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4C5×#6712yescomplemented
415C22×C4C5×S411yescharacteristic, complemented
416C4⋊C4—61no—
417C4⋊C4—62no—
418C4⋊C4—62no—
419C4⋊C4—61no—
420C22⋊C4—64no—
421C22⋊C4—64no—
422C22⋊C4—62no—
423C22⋊C4—64no—
424C22⋊C4—62no—
425C22⋊C4—64no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C3×C5—41noHall
429C22×C3—42no—
430C22×C3—41no—
431C22×C3—81no—
432C22×C3—81no—
433C2×D3—82no—
434C2×D3—82no—
435C2×D3—82no—
436A4C5×#12111yescharacteristic, complemented
437C3×C4—42no—
438C3×C4—81no—
439Dic3—82no—
440C2×C5—22no—
441C2×C5—31no—
442C2×C5—31no—
443C2×C5—31no—
444C2×C5—31no—
445C2×C5—62no—
446C2×C5—242no—
447C2×C5C2×C4⋊S411yescharacteristic
448C2×C5#591211yescharacteristic
449C2×C5C2×GL(2,ℤ/4)11yescharacteristic
450C23—22no—
451C23—31no—
452C23—32no—
453C23—31no—
454C23—31no—
455C23—31no—
456C23—32no—
457C23—32no—
458C23—32no—
459C23—32no—
460C23—31no—
461C23—61no—
462C23—61no—
463C23—61no—
464C23—61no—
465C23—62no—
466C23—61no—
467C23—61no—
468C23—62no—
469C23—62no—
470C23—61no—
471C23—64no—
472C23—62no—
473C23—62no—
474C23—62no—
475C23—62no—
476C23—62no—
477C23—62no—
478C23—62no—
479C23C2×C5×D1211yescharacteristic
480C23C5×#23511yescharacteristic
481C23C2×C5×#6711yescharacteristic
482C23C2×C5×S411yescharacteristic
483D4—122no—
484D4—122no—
485D4—122no—
486D4—122no—
487D4—128no—
488D4—124no—
489D4—124no—
490D4—122no—
491D4—122no—
492C2×C4—22no—
493C2×C4—31no—
494C2×C4—32no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—32no—
498C2×C4—31no—
499C2×C4—62no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—62no—
504C2×C4—64no—
505C2×C4—62no—
506C2×C4—61no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4C2×C5×S411yescharacteristic
517C2×C4C2×C5×S411yescharacteristic
518C2×C3—41no—
519C2×C3—41no—
520C2×C3—41no—
521C2×C3—82no—
522D3—162no—
523C5#8548611yescharacteristic, direct product factor, Sylow
524C22—21no—
525C22—21no—
526C22—31no—
527C22—31no—
528C22—31no—
529C22—31no—
530C22—31no—
531C22—31no—
532C22—31no—
533C22—31no—
534C22—31no—
535C22—34no—
536C22—32no—
537C22—62no—
538C22—62no—
539C22—62no—
540C22—62no—
541C22—61no—
542C22—62no—
543C22—62no—
544C22—62no—
545C22—61no—
546C22—61no—
547C22—122no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22—122no—
553C22—122no—
554C22C5×#93011yescharacteristic, complemented
555C22C5×C4⋊S412yescomplemented
556C22C22×C5×S411yescharacteristic
557C4—21no—
558C4—32no—
559C4—61no—
560C4—122no—
561C4—122no—
562C4—122no—
563C4C5×GL(2,ℤ/4)12yescomplemented
564C3—41noSylow
565C2—22no—
566C2—31no—
567C2—31no—
568C2—31no—
569C2—31no—
570C2—62no—
571C2—242no—
572C2C2×C5×C4⋊S411yescharacteristic
573C2C5×#591211yescharacteristic
574C2C2×C5×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC24×C4×C5D3
socleC24×C5C22×D3
hypercenterC22×C4×C5S4
hypocenterA4C5×#121
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C22×C4×C5

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×C4×C5×A4C2semidirect
2. C2×C5×GL(2,ℤ/4)C2semidirect
3. C2×C5×C4⋊S4C2semidirect
4. C5×(C2×S4)⋊C4C2semidirect
5. C5×C4⋊SL(2,ℤ/4)C2semidirect
16. #85486C5direct (N × H)
17. C24×C4×C5D3semidirect
30. C22×C5×A4D4semidirect
39. C4×C5×A4D4semidirect
43. C22×C4×A4C2×C5semidirect
44. C2×GL(2,ℤ/4)C2×C5semidirect
45. C2×C4⋊S4C2×C5semidirect
46. (C2×S4)⋊C4C2×C5semidirect
47. C4⋊SL(2,ℤ/4)C2×C5semidirect
113. C24×C5D12semidirect
154. C22×C4×C5#67semidirect
155. C22×C4×C5S4semidirect
168. C24×C4C5×D3semidirect
186. C5×A4#121semidirect
192. C22×A4C5×D4semidirect
201. C4×A4C5×D4semidirect
355. C22×C5#930semidirect
356. C22×C5C4⋊S4semidirect
364. C4×C5GL(2,ℤ/4)semidirect
373. C24C5×D12semidirect
414. C22×C4C5×#67semidirect
415. C22×C4C5×S4semidirect
436. A4C5×#121semidirect
523. C5#85486direct (N × H)
554. C22C5×#930semidirect
555. C22C5×C4⋊S4semidirect
563. C4C5×GL(2,ℤ/4)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C4×C5×A42
C2×C5×GL(2,ℤ/4)2
C2×C5×C4⋊S42
C5×(C2×S4)⋊C42
C5×C4⋊SL(2,ℤ/4)2
C5×#28473
C5×#9304
#854865